秩度量码密度的几何方法
A geometric approach to the density of rank-metric codes
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中文总结 AI 辅助
本文通过有限域上射影簇的几何方法研究秩度量码的渐近密度，利用行列式簇和拟自反性推广，结合Chebotarev密度定理，确定了先前未知的极限情形。
中文摘要 AI 辅助
我们研究了有限域上几何不可约射影簇的$\mathbb{F}_q$-无点线性截面的渐近密度。随后，我们通过行列式簇将这些结果应用于秩度量码。我们的方法恢复了密度趋于$0$或$1$的已知情形，并确定了先前未知情形下的极限。为计算这些先前未知的极限，我们将拟自反性的概念推广到高维簇，并证明行列式簇满足该性质。这使我们能够引用有限域上簇的Chebotarev密度定理，从而获得所需的估计。
英文摘要
We study the asymptotic density of $\mathbb{F}_q$-point-free linear sections of geometrically irreducible projective varieties over finite fields. We then apply these results to rank-metric codes via determinantal varieties. Our approach recovers the known cases in which the density tends to $0$ or $1$ and determines the limit in the cases where it was previously unknown. To compute these previously unknown limits, we extend the notion of quasireflexivity to higher-dimensional varieties and show that determinantal varieties satisfy this property. This allows us to invoke the Chebotarev density theorem for varieties over finite fields to obtain the desired estimate.
